The Black-Scholes Equation and Volatility...Black-Scholes equation and numerical simulations of the Black-Scholes partial differ-ential equation, using different values for volatility.

The Black-Scholes Equation and Volatility

Samara LaliberteDept. of Mathematics

UMass DartmouthDartmouth MA 02747

Email: [email protected]

The Black Scholes formula has often been described as using ”the wrong number in the wrongformula to get the right price”

Abstract

The Black-Scholes equation is a commonly used model in financial mathematicswhich allows one to analyze the price or worth of a derivative over time and changesin underlying assets. A major issue with equation is that it typically assumes a morestable volatility than what actually is. In this paper, I will discuss the formation of theBlack-Scholes equation and numerical simulations of the Black-Scholes partial differ-ential equation, using different values for volatility.

1 History

In 1973 Fischer Black, a finance contractor and Myron Scholes, an assistant professor of financeat MIT published a paper explaining the Black-Scholes Option Pricing Model. Almost all presentday approaches and techniques used to estimate pricing are based on this Black-Scholes Model.Scholes received the Noble Prize in Economics in 1997, with an mention to Black because he hadpassed away in 1995.

2 Derivative

A derivative is an agreement between two parties that has a value. This value is based on the priceof an underlying asset, based on it’s past behavior and future predictions. The underlying asset canbe anything that has a value such as a stock, bond, currency or interest rate. The values of theseassets will change but the agreed value of the derivative will not.

3 Option

An option is a derivative that creates an agreement between two parties on a price and amount oftime the parties have to buy/sell the asset. The buyer may by this option up until the expiration

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date at the agreed price, but does not have to. On the other hand the seller must sell at any time thebuyer decides to exercise his right to buy.

3.1 American Option

May be exercised any date before or on the expiration date. This makes the Black-Scholes Equationless accurate in price prediction because it only predicts the price at the time of expiration.

3.2 European Option

May only be exercised on the expiration date. The Black-Scholes equation can predict the pric-ing of a European option with better accuracy because of the conditions to only purchase on theexpiration date.

4 The Black-Scholes Equation

The Black-Scholes Equation is an equation used in financial mathematics. It calculates the priceor worth of a derivative over a certain amount of time. The price represents the price at the timeof expiration of the option, but not at anytime before. This allows the equation to calculate largenumbers of option prices in a short amount of time. This is also one of the major flaws of theequation considering in the United States one can exercise the right to buy an option at anytimeuntil and including the expiration date. Another major flaw of the Black-Scholes equation is theuse of constant volatility. Volatility is the measure of risk based on the standard deviation of thevalue of an asset over a specific time, in this case until the expiration. In reality the value of an assetwill change randomly, not with a specific constant pattern. Despite these flaws the Black-Scholesequation is an important tool in finance.

5 Portfolio

A portfolio is the collection of investments held by a institution or individual. These investmentsinclude all types of assets from a bank account to bond. The purpose of holding a portfolio is tominimize the risk of owning many assets at one time.

dΠ = ∆tdSt − dP (St, t) (1)

• Π = The portfolio

• St = Price of Stock

• ∆t = number of shares owned at time t

• P (St, t) = Price of the option

This equation represents the derivative of the equation of a portfolio, or the change in the value ofthe portfolio over time. By finding values for dSt, the change in the price of the stock over time

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and dP (St, t), the change in the price of the option over time. An equation for the change in thevalue of the portfolio can be formed. Using and equation for Geometric Brownian Motion andIto’s Lemma values are found.

6 Geometric Brownian Motion

One requirement for the Black-Scholes equation to hold is an assumption that the investment beingmade is risk free. A risk free investment is one that has a value that does not vary much from theoriginal value over the amount of time it is being held. Using Geometric Brownian Motion the riskcan be eliminated.

6.1 Brownian Motion

Brownian Motion is a stochastic process. A stochastic process is one that happens at random,there is no real prediction for the behavior of it over time. The definition of Brownian Motionis the prediction that particles suspended in fluid will move at random. This arbitrary pattern isconsidered similar to the way prices of stocks in the stock market change over time.

Brownian Motion

This is a graph of Brownian Motion: the random movement of a particle in fluid.

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Price of Cisco Stock

This graph is an example of a stock price changing over time. It looks similar to the graph ofBrownian Motion but a little less sporadic. There is no accurate way to predict an exact stock price,Brownian Motion is used as component the Black-Scholes equation to show this.

6.2 Equation for Geometric Brownian Motion

dSt = µStdt + σStdzt (2)

• µ = Rate at which value of stochastic process changes, annualized

• σ = Volatility

• dzt = The risk

This equation represents the change in the price of the stock over a certain amount of time: dSt.This is one of the variables needed in the equation for the derivative of the portfolio. This equationalso introduces the risk term: dzt. Eventually by eliminating this term the portfolio will be riskfree, a condition for the Black-Scholes equation.

7 Ito’s Lemma

Ito’s lemma is part of Ito calculus which allowes the of use calculus in stochastic processes. Itocalculus can be used to find integrals but in this case a part of Itos calculus, Ito’s lemma, will beused to find the derivative of P (St, t). Ito’s lemma is the stochastic calculus version of the chainrule, using the taylor series expansion. It is commonly used in financial mathematics, specificallyin the derivation of the Black-Scholes equation.

Using Ito’s Lemma on dP (St, t) forms the equation:

dP (St, t) = (∂P (St, t)

∂SµSt +

∂P (St, t)

∂t+

1

2

∂2P (St, t)

∂S2σ2S2

t ) (3)

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8 Equation for the Derivative of a Portfolio

Geometric Brownian Motion found a value for the derivative of the stock price and using Ito’slemma a value for the price of the option was found. By substituting these values into the equationfor the derivative of the portfolio a new equation is formed.

dΠ = (∆tµSt −∂P (St, t)

∂SµSt −

∂P (St, t)

∂t− 1

2

∂2P (St, t)

∂S2σ2S2

t )dt+ (∆tσSt −∂P (St, t)

∂SσSt)dzt (4)

Now the dzt or the risk term is included in the equation. The Black Scholes equation requiresthat there be no risk, so this term must be eliminated by finding a ∆t or a number of shared ownedat time t that gets rid of the risk. The number of shares which does this is:

∆t =∂P (St, t)

(∂S)(5)

9 Risk Free Portfolio

The definition of a risk free portfolio is one that does not change in value over the time it is held.By eliminating the dzt term the risk is eliminated but a interest rate must still be factored in. To dothat we must take:

dΠ = (−∂P (St, t)

∂SSt − P (St, t))dt (6)

and set it equal to the equation with a risk free interest rate r. By setting dΠ = rΠdt the newequation with an interest rate is created.

rP (St, t) =∂P (St, t)

∂t+ rSt

∂P (St, t)

∂S+

1

2σ2S2

t

∂2P (St, t)

∂S2(7)

This is the Black-Scholes partial differential equation.

10 Volatility

”It (volatility) is only a good measure of risk if you feel that being rich then being poor is thesame as being poor then rich” - Peter Carr(Managing Director at Morgan Stanley)

Volatility is the variation of the price of an asset over a certain amount of time, the time untilexpiration. The Volatility does not measure whether the prices increase or decrease it merely shows

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how large or small these price changes were or can be expected to be. The Black Scholes modelassumes this value to be constant, which is not accurate to real markets. In reality volatility changesfrom high to low many times until expiration. An asset experiences high volatility when the pricesmove up and down frequently. Low volatility is when the prices do not move much over time.

10.1 Implied Volatility

Implied volatility is volatility calculated from the price of a derivative. It is used in pricing topredict a price for a derivative using current, past or future prices.

10.2 Historical Volatility

Historical Volatility is a prediction of the value of a derivative using prices from the past.

This is an example of a Black Scholes option pricing model using constant volatility. The graphhas a constant slope which is not accurate to a volatility surface representing a real market.

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This graph shows what a volatility surface should look like, the shape of a smile or skew

This is an example of how volatility changes over time. The jumps are frequent but rarely go over.03. This is very different than what a Black-Scholes plot of volatility would look like, a straight

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line at a certain y value. Although the difference is small, there is a difference and when investinghard earned money, it is important to use a value accurate to real markets.

11 Code

clear all; close all; clc

r = log(1.04)/365; % bond growth ratemu = log(1.05)/365; %Stock price growth rateSigma = .01; %Stock price volatility, using random valuesinit = 100; % initial stock and bond pricec = 95; % exercise/strike price

N=500; % number of steps to takeT=180; % expiration timeh=T/N; % time stept=(0:h:T); % t is the vector [0 1h 2h 3h ... Nh]

% Initial values

Finding initial values for the equation for the price of an option, number of shares of stock andnumber of shares of a bond

b(1)=init; % initial bond price B_1p(1)=init; % initial stock price S_1

s = T; % time argument - Changing variable T to sd1 = (log(p(1)/c)+(r+(sigmaˆ2)/2)*s) ./ (sigma*sqrt(s));d2 = d1 - sigma*sqrt(s);

d1 and d2 are variables used in solving the Black-Scholes equation

x(1) = p(1) .* normcdf(d1) - c*exp(-r*s) .* normcdf(d2); %price of option

x(1) is equal to the equation for the price of an option, also represented as C(St, t) where..

C(St, t) = N(d1)S1 −N(d2)Ke−r(T−t) (8)

• p(1) = S1 = the initial stock price

• normcdf = N =normal cumulative distribution function. The cumulative distribution func-tion of the normal distribution. The normal distribution is a probability distribution used todescribe random variables which cluster to a certain mean value. The cumulative distributionfunction shows the probability that a random variable will be less than this value.

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• N(d2) - Probability that the stock price will be above K

• K = c = Strike price, agreed price of derivative

• r = r = risk free interest rate

• T − t = s = time to expiration

y(1) = x(1);n(1) = normcdf(d1)+(normpdf(d1)-c*exp(-r*s)*normpdf(d2)/p(1))/(sigma*sqrt(s)); %shares of stock

Shares of Stock = Nd1 +(Np(d1)−Ke−r(T−t) ∗ Np(d2)

S1)

σ(√T − t)

(9)

• normpdf= Np = Normal probability density function, probability of this random variable oc-curring at a given point.

m(1) = (x(1)-n(1)*p(1))/b(1) % shares of bond

Shares of bond =(C(S1, t)−Nd1 +

(Np(d1)−Ke−r(T−t)∗Np(d2)

S1)

σ(√T−t) ∗ S1)

B1

(10)

m(1)= shares of the bond = (Price of the option - shares of stock * initial stock price) / the initialbond price.Finding values for the functions that change over time by taking steps:

for i=1:N % start taking stepsb(i+1)=b(i)+r*b(i); % bond price:

Bond price + risk free interest rate multiplied by bond price = total bond price, bond price afterinterest rate is included over time... summed from i = 1 : N . Adding the interest rate to thesolution.

p(i+1)=p(i)+mu*p(i)*h+sigma*p(i)*sqrt(h)*randn; %stock price

Stock Price using i from 1 to N steps...

p(i+ 1) = Si + µSih+ σSi√

(h)(randn) (11)

• µ is the Drift rate or the change in the average value of the stochastic process

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• h is the time step T/N... T= expiration time and N = number of steps

• σ is the volatility

• randn= gives pseudorandom values drawn from the standard normal distribution

y(i+1) = y(i) + n(i)*(p(i+1)-p(i)) + m(i)*(b(i+1)-b(i)); % wealth

• This represents how much the options price will grow or shrink, the wealth over time.

• By adding the change in price of the stock and the change in price of the bond

• Finding the change in price of option by adding the changes of stock price and bond price

• Using Newton’s method to find the change in value or derivative of the stock price and bondprice over values of i from 1 to N .

s = T-t(i+1); % time argument using steps

d1 = (log(p(i+1)/c)+(r+(sigmaˆ2)/2)*s) ./ (sigma*sqrt(s));d2 = d1 - sigma*sqrt(s);

d1 and d2 using p(i+ 1) instead of p(1), used in taking stepsUse changing time, s and i + 1 in the original equations to get change in shares of stocks andbonds, and price of the option over time. Re-stating values of x(i), n(i), m(i), using the stepsfrom i = 1 : N

x(i+1) = p(i+1) .* normcdf(d1) - c*exp(-r*s) .* normcdf(d2);%price of option

Equation for the price of an option using time steps, and a changing initial value for the stock price.

n(i+1) = normcdf(d1) + (normpdf(d1)-c*exp(-r*s)*normpdf(d2)/p(i+1))/(sigma*sqrt(s)); %Shares of stock

Equation for the shares of the stock using time steps and change in initial prices over time.

m(i+1) = (x(i+1)-n(i+1)*p(i+1))/b(i+1); % shares of bond

(Price of the option over N steps - shares of the stock over N steps* Price of stock over N steps) /Bond price = the number of shares of a bond

end;

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Graphing surface and grid. Use tt, xx and uu to make surface:

[xx,tt] = meshgrid(60:1:120,0.05:3:180); % prepare points on a grid

gg = (log(xx/c)+(r+(sigmaˆ2)/2)*tt) ./ (sigma*sqrt(tt));

gg =(log(xx

k) + (r + σ2

2)tt)

σ√

(tt)(12)

Taking d1 and replacing the variable for stock price with xx and the variable for time to tt

uu = xx .* normcdf(gg) - c*exp(-r*tt) .* normcdf(gg - sigma*sqrt(tt));

uu = xxN(gg)−Ke−rttN(gg − σ√

(tt) (13)

This is the equation for the price of an option using gg in place of d1 and again replacing thevariable for stock price with xx and the time argument with ttuu becomes the 3-D surface, showing the change in price of the option.Then creating a graph, label and setting axis’ :

mesh(tt,xx,uu);shading faceted;Labelingxlabel(’Time remaining’);ylabel(’Stock price’);zlabel(’Wealth’);title(’Wealth of Black Scholes Portfolio’);axis([0 180 60 120 0 100]); %Set size of axisgrid on;hold on;

plot3(T-t, p, x+0.1);

Plot line in 3d space: price of an option of time until expiration in steps of x +.01....blue line

pauseclfend

12 Low Volatility

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The surface has a somewhat constant slope signifying a remotely constant change in stock pricesover time. The stock prices have semi-regular jumps from high to low over the 180 days becausethe volatility is set at a small value, .01.

13 High Volatility

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The volatility surface shown here has a greater slope due to the larger volatility. The stock pricesare more erratic because of the size of the volatility which is set at .03.

14 Personal Reflection

I learned a lot about financial mathematics this semester. There were a few problems I ran intowhich slowed me down and didn’t allow me to accomplish all I hoped to. The first set back wasactually understanding the Black-Scholes equation. I had underestimated how complicated justfiguring out what the equation did and how it was formed would be. I was hoping to be able tounderstand the solution and create a code of my own rather than having to alter someone else’s.With that, I did learn in depth how the Black-Scholes equation is formed and actually understand

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it to a point where I could probably explain it to someone. That I think is an accomplishmentin itself. I’m honestly more interested in pure mathematics than computational so I was hopingto understand how to solve the equation without using a code. I tried for a while to do this, butsort of gave up in order to focus more on volatility and why it is important to the Black-Scholesequation. I hope to explore the solution more in depth in the future. My original goal in all thiswas to find a better way to computationally compute volatility in the Black-Scholes equation. Ididn’t realize in the beginning that there were many different models for volatility and it wasn’tjust a somewhat simple equation. Volatility became a whole new ”Black-Scholes equation” to me.While trying to produce different graphs and models of the Black-Scholes equation and volatilityI ran into the issue of what to compare my future results to. There is no way to find a completelyaccurate price or volatility until it is observed. In the future I think I will compare my results tovalues form the past in order to understand the error. Because of my lack of skill in MATLAB Iwasn’t able to incorporate that equation into my code, in order to make a changing volatility. ThatI will consider my next goal in CSUMS. Overall I was not prepared for how complex and newfinancial mathematics would be to me. There are many terms and processes on must understandbefore getting into the computation. I have to say I learned a lot about this area of mathematicsand am very interested in learning more. I hope to expand my ability to alter codes and hopefullycreate my own from scratch using the Heston model for volatility. I Though out this semesterLATEX has become second nature to me, even though I always have so many errors I can producea semi-goodlooking paper or beamer presentation as if I was typing something up on WORD.MATLAB has always been a challenge for me, something in my brain just doesn’t get that and Ididn’t progress as much in this area as I would have liked to this semester because of all the timespent just building the equation. Another challenge which does not necessarily have to do withmath itself is my fear of public speaking. Even though I’m still horrified at getting up in front ofjust a couple people, I got some experience under my belt. Despite there not being many peopleat Amherst when I presented I went there intending to talk in front of as many people showed andsucceeded in that goal. I look back and even though I didn’t accomplish everything I would haveliked to I feel as if I improved my skills in many areas and found a topic I am very interested incontinuing researching.

15 CSUMS Review

CSUMS is a good experience which allowed me to explore something I am actually interested in.The first semester I was given a topic which I thought was a very good strategy. Coming intoCSUMS I was somewhat scared that I wouldn’t come up with a research topic and really had noidea how to do research in math. The fact that a paper was given to me and I could just dissect itand try to understand the little pieces which seemed impossible was fun to me. Eventually I gainedan understand of fourier series and with my partner created a code, this gave me confidence fornext semester. This semester I was able to pick my own research topic which is one thing I likeabout CSUMS. In every other class you are told what you have to learn, study and memorize fora test. Here I could actually enjoy the topic I was learning about and pick and choose what piecesI’d like to learn and what I would save for later.

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16 References

• Financial Numerical Recipes in C++, Bernt Arne Odegaard

• An introduction to Computational Finance, Omur Ugur

• Explanation and Derivation of the Black-Scholes PDE, Doug Vestal

• Black-Scholes model of option pricing, upenn.edu

• http://financial-dictionary.thefreedictionary.com

• Hoadley Training and Investment Tools-www.hoadley.net

• investopedia.com

• www.mathworks.com

• http://www-math.bgsu.edu/ zirbel/sde/, Bowling Green State Universtiy

17 Acknoledgements

Advisor Sigal Gottlieb and Sidafa Conde

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